Who Stole the Tarts? Part X

Shortly after the cookbook was returned to the Queen, it was stolen a second time—again by either the Duchess, the Cook, or the Cheshire Cat.

At the trial they made exactly the same statements as at the last trial. Only this time, the thief lied and the other two either both lied or both told the truth.

Who stole the cookbook this time?

(Source: Alice in Puzzle Land: A Carrollian Tale for Children Under Eighty by Raymond Smullyan)

Recall that these were the statements from the last trial:

"The Cheshire Cat stole it!" said the Duchess at the trial.
"Oh, yes, I stole it!" said the Cheshire Cat with a grin.
"I didn't steal it!" said the Cook.

The most straightforward way is to consider whether each individual in turn might be the thief.

If the Duchess stole the cookbook, then her statement that the Cheshire Cat stole it is false, which is good so far because it is required that the thief lied. Then the Cheshire Cat is also lying because he claimed to have stolen it. But the Cook is telling the truth when she says she didn't steal it, and it is required that the two non-thieves either both told the truth or both lied. So the Duchess is not the thief.

If the Cheshire Cat stole the cookbook, then the other two are both telling the truth, which is good. But because it is required that the thief lied, the Cheshire Cat would have had to say he did not steal the cookbook. The Cheshire Cat is now also exculpated.

The only remaining option is the Cook. But to verify, it checks out that she lied, because she said she didn't steal it. And everyone else here has also perjured themselves so the criterion that the others must either both lie or both tell the truth is also met.